Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

Global Error Minimization method for solving strongly nonlinear oscillator differential equations

A modified variational approach called Global Error Minimization (GEM) method is developed for obtaining an approximate closed-form analytical solution for nonlinear oscillator differential equations. The proposed method converts the nonlinear differential equation to an equivalent minimization problem. A trial solution is selected with unknown parameters. Next, the GEM method is used to solve the minimization problem and to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations (ODEs). This approach is simple, accurate and straightforward to use in identifying the solution. To illustrate the effectiveness and convenience of the suggested procedure, a cubic Duffing equation with strong nonlinearity is considered. Comparisons are made between results obtained by the proposed GEM method, the exact solution and results from five recently published methods for addressing Duffing oscillators. The maximal relative error for the frequency obtained by the GEM method compared with the exact solution is 0.0004%, which indicates the remarkable precision of the GEM method.     

基于LS-SVM求解非线性常微分方程组的近似解

提出了一种基于最小二乘支持向量机(LS-SVM)的改进方法求解非线性常微分方程组初值问题的近似解.利用径向基核函数(RBF)可导的特点对LS-SVM模型进行改进,将含核函数导数形式的LS-SVM模型转化为优化问题进行求解.方法可在原始对偶集中获得近似解的最佳表示,所得近似解连续可微,且精度较高.给出数值算例,通过与真实解的对比验证了所提方法的准确性和有效性.     

High-order continuous third derivative formulas with block extensions for y″=f(x,y, y′)

A class of high-order continuous third derivative formulas (CTDFs) for second-order ordinary differential equations (ODEs) is proposed. The CTDFs are used to generate main and additional methods which are combined to give the block third derivative formulas (BTDFs). The BTDFs are applied as single-block matrix equations to simultaneously provide approximate solutions for second-order ODEs. The stability properties of the BTDFs are discussed. Numerical examples are given to illustrate the accuracy and efficiency of the block extensions of the methods.     

Exact linearization of nonlinear ordinary autonomous differential equations

Many methods for finding exact solutions to nonlinear ordinary differential equations (ODE) are based on certain euristic rules. The author suggested a newexact linearization method that provides an algorithmic procedure for constructing exact solutions for some important classes of ODEs [1].     

Exact and approximate Riemann solvers for compressible two-phase flows

Numerical methods for solving equations of two-phase hydrodynamics, which describe the flow of a dispersed solid and gas mixture are considered. The Godunov method is applied as the main approach to approximate numerical fluxes in solutions of the relevant Riemann problems. The formulations of these problems for the solid and gas phases are given, their exact analytical solution is described, and possible simplified approximate solutions are discussed. The obtained theoretical results are applied to the construction of a discrete model, which results in the generalization of the well-known Godunov-type and Rusanov-type methods to the case of nonequilibrium two-phase media. The numerical results involve the verification of the constructed methods on the analytical solutions of two-phase equations.     

Hermite–Sobolev orthogonal functions and spectral methods for second- and fourth-order problems on unbounded domains

Hermite spectral methods using Sobolev orthogonal/biorthogonal basis functions for solving second and fourth-order differential equations on unbounded domains are proposed. Some Hermite–Sobolev orthogonal/biorthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. The convergence is analyzed and some numerical results are presented to illustrate the effectiveness and the spectral accuracy of this approach.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

Solving differential equations with Fourier series and Evolution Strategies

A novel mesh-free approach for solving differential equations based on Evolution Strategies (ESs) is presented. Any structure is assumed in the equations making the process general and suitable for linear and nonlinear ordinary and partial differential equations (ODEs and PDEs), as well as systems of ordinary differential equations (SODEs). Candidate solutions are expressed as partial sums of Fourier series. Taking advantage of the decreasing absolute value of the harmonic coefficients with the harmonic order, several ES steps are performed. Harmonic coefficients are taken into account one by one starting with the lower order ones. Experimental results are reported on several problems extracted from the literature to illustrate the potential of the proposed approach. Two cases (an initial value problem and a boundary condition problem) have been solved using numerical methods and a quantitative comparative is performed. In terms of accuracy and storing requirements the proposed approach outperforms the numerical algorithm.     

On Variational and PDE‐Based Distance Function Approximations

In this paper, we deal with the problem of computing the distance to a surface (a curve in two dimensional) and consider several distance function approximation methods which are based on solving partial differential equations (PDEs) and finding solutions to variational problems. In particular, we deal with distance function estimation methods related to the Poisson‐like equations and generalized double‐layer potentials. Our numerical experiments are backed by novel theoretical results and demonstrate efficiency of the considered PDE‐based distance function approximations.     

Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation

This paper proposes three fast and high accuracy numerical methods for solving a nonlinear partial differential equation (PDE) describing water waves and called the Boussinesq (Bq) equation. We numerically solve the Bq equation with fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original PDE with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) which will be solved with fourth-order time-stepping methods. After transforming the equation to a system of ODEs, the linear operator is not diagonal, but we can implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented. Also we investigate the conservation of mass for Bq equation.     

A localized approach for the method of approximate particular solutions

The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems.In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

Direct solutions of nth-order IVPs by homotopy-perturbation method

In this paper, a class of linear and nonlinear nth-order initial value problems (IVPs) is considered. The solutions of these IVPs are obtained by the homotopy-perturbation method (HPM). The HPM can be considered as one of the new methods belonging to the general classification of perturbation methods. Generally, the HPM deals with exact solvers for linear differential equations and approximative solvers for nonlinear equations. Several test cases are chosen to demonstrate the efficiency of HPM.     

Error bounds for initial value problems by optimization

The computation of the error bounds for approximate solutions of initial value problems for ordinary differential equations has a long and successful history. This paper presents a new scheme to compute such bounds with uncertain initial conditions using preconditioned defect estimates and optimization techniques. These bounds are based on the newly developed concept of conditional differential inequalities. The scheme is implemented in MATLAB and AMPL. The resulting enclosures are compared with the packages VALENCIA-IVP, VNODE-LP and VSPODE for bounding solutions of ODEs. The current prototype uses heuristics to solve the global optimization subproblems. Hence the bounds obtained in the numerical experiments are not fully rigorous. The latter can be achieved by using rigorous global optimization and rounding error control, but the effect on the bounds is likely to be marginal only.     

Solving the mesh-partitioning problem with an ant-colony algorithm

Many real-world engineering problems can be expressed in terms of partial differential equations and solved by using the finite-element method, which is usually parallelised, i.e. the mesh is divided among several processors. To achieve high parallel efficiency it is important that the mesh is partitioned in such a way that workloads are well balanced and interprocessor communication is minimised. In this paper we present an enhancement of a technique that uses a nature-inspired metaheuristic approach to achieve higher-quality partitions. The so-called multilevel ant-colony algorithm, which is a relatively new metaheuristic search technique for solving optimisation problems, was applied and studied, and the possible parallelisation of this algorithm is discussed. The multilevel ant-colony algorithm performed very well and is superior to classical k-METIS and Chaco algorithms; it is even comparable with the combined evolutionary/multilevel scheme used in the JOSTLE evolutionary algorithm and returned solutions that are better than the currently available solutions in the Graph Partitioning Archive.     

A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization

A general algorithm is presented to approximately solve a great variety of linear and nonlinear ordinary differential equations (ODEs) independent of their form, order, and given conditions. The ODEs are formulated as optimization problem. Some basic fundamentals from different areas of mathematics are coupled with each other to effectively cope with the propounded problem. The Fourier series expansion, calculus of variation, and particle swarm optimization (PSO) are employed in the formulation of the problem. Both boundary value problems (BVPs) and initial value problems (IVPs) are treated in the same way. Boundary and initial conditions are both modeled as constraints of the optimization problem. The constraints are imposed through the penalty function strategy. The penalty function in cooperation with weighted-residual functional constitutes fitness function which is central concept in evolutionary algorithms. The robust metaheuristic optimization technique of the PSO is employed to find the solution of the extended variational problem. Finally, illustrative examples demonstrate practicality and efficiency of the presented algorithm as well as its wide operational domain.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Solving biharmonic equation using the localized method of approximate particular solutions

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.